A knot in $S^3$ is rationally slice if it bounds a disk in a rational homology ball. We give an infinite family of rationally slice knots that are linearly independent in the knot concordance group. In particular, our examples are all infinite order. All previously known examples of rationally slice knots were order two. The proof relies on bordered and involutive Heegaard Floer homology. This is joint work with Sungkyung Kang, JungHwan Park, and Matt Stoffregen.